If you don't know what you are doing it may not be a good idea to dremel your case, but it is not too hard.
1. First of all you could sell you current case as you mentioned , and buy a big full tower case. This is the expensive option, but if you decide to cut your HDD cage you will have to buy a dremel. This would solve your problems.
2. You could sell your windforce x3 card and get a two reference gtx670. I saw a thread a while ago that said that the temperatures of two gtx670 cards with windforce coolers are not so good. They are great cards (I have one myself), but they exhaust hot air into the case, and if you have two it would exhaust all the air onto the other card. They are usually best for a single GPU setup. With a reference card the hot air is exhausted through the vents in the PCI-e bracket which is better when you have two cards in SLI.. So two of them in SLI would be an alternative to buying a new case.
Personally I wouldn't like having two mismatched cards, and as they would have different coolers (one exhausting air into case, one exhausting out of the case) you would have to have very good case ventilation. You could have the reference card above the windforce card (exhausting hot air out of the case), and the windforce card on the bottom. That way you give the windforce cooler room to do its job.
Or if you are very CAREFUL then you could take out the things from the case and cut into the drive cage. But I would not do it.